A dc motor would work, an ac motor would be cheaper if you can run at a fixed speed, something like a paravalux geared motor is what I'd be looking for, if you want variable speed then the cost of a variable speed ac drive would probably make an ac motor prohibitive a dc motor speed control is easy to make and cheap, or a universal motor for that matter.
You could use a standard motor starter to do the trick, pull one of the wires off the stop button and then wire it in series with the normally closed contact of a button actuated limit switch, run an arm with a rubebr roller on the end (to protect damaging the wire) to make contact with the wire (and trigger the switch) when one reel is full and another switch for when the other reel is empty, connect both of the swtich contacts in series so whene one of them is actuated is has the same effect as pressing the stop button on the starter.
Another way is to get hold of a turns counter with a preset contact, you preset the no. of turns and the contact operates at that no.
You dont need a micrcontroller for this.